Question 12.4: Testing the Correlation Coefficient Refer to the management ......

The same five steps of hypothesis testing are used. The test statistic is t-stat.

Step 1: Define the null and alternative hypotheses

H_0: ρ = 0 (newspaper advertisements and TV sales are not related)

H_1: ρ ≠ 0 (newspaper advertisements and TV sales are related)

This is a two-tailed hypothesis test that shows no relationship in the null hypothesis.

Note: The closer r is to zero, the more likely it is that the null hypothesis will be accepted.

Step 2: Determine the region of acceptance of the null hypothesis

The test is based on the t statistic (as used in chapters 7 and 8). To read off the critical t-limits for the region of acceptance, both a level of significance and degrees of freedom for the test are required.

Given α = 0.05. For a simple regression equation:

Degrees of freedom (df) = n − 2          12.10

In this example, df = 12 − 2 = 10. Then t-crit = ±2.228 (Table 2, Appendix 1). Thus the region of acceptance for H_0 is −2.228 ≤ t ≤ +2.228.

The decision rule is then stated as follows:

• Accept H_0 if t-stat lies between −2.228 and +2.228 inclusive.
• Reject H_0 if t-stat lies below −2.228 or above +2.228.

Step 3: Calculate the sample test statistic (t-stat)

The sample test statistic is t-stat, which is calculated using the following formula:

t-stat = r\sqrt{\frac{(n-2)}{1-r^2} }          12.11

In the example, r = 0. 8198 and n = 12. Then:

t-stat = 0.8198\sqrt{\frac{(12-2) }{(1-0.8198^2)}} = 0.8198 × \sqrt{\frac{10}{0.3279} } = 4.527

The p-value can be found using T.DIST.2T(t-stat,df = n – 2)

i.e. p-value = T.DIST.2T(4.527,10) = 0.0011

Step 4: Compare the sample test statistic to the region of acceptance

The sample test statistic, t-stat = 4.527 lies outside (well above) the region of acceptance of H_0, as shown in Figure 12.19.

Step 5: Draw statistical and management conclusions

Since t-stat > + t-crit (and p-value << α), we reject H_0 at the 5% level of significance. There is strong enough sample evidence to conclude that the population correlation coefficient is not zero. The alternative hypothesis is probably true.

From a management viewpoint, the sample evidence indicates that there is a genuine strong positive statistical relationship between the number of advertisements placed (x) and the weekly flat-screen TV sales (y) in the population.